"Equilateral" already states that the sides are all equal.
"Equiangular" already states that the angles are all equal.

Are these acceptable?. . The opposite sides are parallel (also true for a parallelogram).. . The diagonals are equal (also true for a rectangle).. . The diagonals are perpendicular (also true for a rhombus).. . The diagonals bisect each other (also true for a parallelogram).